arXiv:1502.00322v1 [quant

arXiv:1502.00322v1 [quant-ph] 1 Feb 2015
An extended Dirac equation in noncommutative
space-time
R. Vilela Mendes∗†
Centro de Matem´atica e Aplica¸c˜oes Fundamentais, Univ. Lisboa
Av. Gama Pinto 2, 1649-003 Lisboa, Portugal
Abstract
Stabilizing, by deformation, the algebra of relativistic quantum mechanics a non-commutative space-time geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation
which has both a massless and a large mass solution. The nature of the
solutions is discussed, as well as the effects of coupling the two solutions.
1
The stable Heisenberg-Poincar´
e algebra and
noncommutative space-time
When models are constructed for the natural world, it is reasonable to expect
that only those properties of the models that are robust have a chance to be
observed. Models or theories being approximations to the natural world, it is
unlikely that properties that are too sensitive to small changes (that is, that
depend in a critical manner on particular values of the parameters) will be well
described in the model. If a fine tuning of the parameters is needed to reproduce
some natural phenomenon, then the model is basically unsound and its other
predictions expected to be unreliable. For this reason it would seem that a good
methodological point of view, in the construction of physical theories, would be
to focus on the robust properties of the models or, equivalently, to consider only
models which are stable, in the sense that they do not change, in a qualitative
manner, when some parameter changes.
This point of view had a large impact in the field of non-linear dynamics,
where it led to the rigorous notion of structural stability [1] [2]. As pointed out by
Flato [3] and Faddeev [4] the same pattern seems to occur in the fundamental
theories of Nature. In fact, the two physical revolutions of the last century,
namely the passage from non-relativistic to relativistic and from classical to
quantum mechanics are deformations of two unstable Lie algebras to two stable
∗ also
at Instituto de Plasmas e Fus˜
ao Nuclear - IST
[email protected]
† [email protected],
1
ones. A mathematical structure is said to be stable (or rigid ) for a class of
deformations, if any deformation in this class leads to an equivalent (isomorphic)
structure. When going from Galilean to relativistic mechanics, the Galilean
algebra, an isolated point, is deformed to the stable Lorentz algebra and on the
transition from classical to quantum mechanics the unstable Poisson algebra is
deformed to the stable Moyal algebra.
This situation motivated the question of whether the full algebra of relativistic quantum mechanics, the Heisenberg-Poincar´e algebra, would itself be stable.
The answer was that it is not and that one possible deformation to a stable one
is defined by the following commutators [5]:
[Mµν , Mρσ ]
[Mµν , Pλ ]
[Mµν , xλ ]
[Pµ , Pν ]
[xµ , xν ]
[Pµ , xν ]
[Pµ , ℑ]
[xµ , ℑ]
=
=
=
=
=
=
=
=
i(Mµσ ηνρ + Mνρ ηµσ − Mνσ ηµρ − Mµρ ηνσ )
i(Pµ ηνλ − Pν ηµλ )
i(xµ ηνλ − xν ηµλ )
−i Rǫ42 Mµν
−iǫ5 ℓ2 Mµν
iηµν ℑ
−i Rǫ42 xµ
iǫ5 ℓ2 Pµ
(1)
In this algebra, that will be denoted ℜℓ,R = {Mµν , Pµ , xµ , ℑ}, Mµν are the
generators of the Lorentz group, Pµ and xµ the momenta and coordinates, ℑ a
non-trivial operator that replaces the center of the Heisenberg algebra and ǫ4 , ǫ5
are ± signs. Velocities and actions are in units of c and ~ (that is c = ~ = 1). The
stable algebra ℜℓ,R is isomorphic to the algebra of the 6−dimensional pseudoorthogonal group with metric
ηaa = (1, −1, −1, −1, ǫ4, ǫ5 ), ǫ4 , ǫ5 = ±1
(2)
The nonvanishing right hand side of the [Pµ , Pν ] commutator simply means
that flat space is an isolated point in the set of arbitrarily curved spaces. This
is why Faddeev [4] points out that Einstein’s theory of gravity may also be
considered as a deformation in a stable direction. Einstein’s theory is based on
curved pseudo Riemann manifolds. Therefore, in the set of Riemann spaces,
Minkowski space is a kind of degeneracy whereas a generic Riemann manifold is
stable in the sense that in its neighborhood all spaces are curved. However, as
long as one is concerned with the kinematical group of the tangent space to the
space-time manifold, and not with the group of motions in the manifold itself, it
is perfectly consistent to take R → ∞ and this deformation would be removed.
In particular, because the curvature is not a constant, R cannot have the status
of a fundamental constant.
In contrast, for the other stabilizing deformation, associated to the ℓ constant, there is no obvious reason to remove it and ℓ might play the role of a new
fundamental constant. Taking the ℜℓ,∞ algebra as the kinematical algebra of
tangent space, the main features are the non-commutativity of the xµ coordinates and the fact that ℑ, previously a trivial center of the Heisenberg algebra,
2
becomes now a non-trivial operator. These are however the minimal changes
that seem to be required if stability of the algebra of observables (in the tangent
space) is a good guiding principle. Two constants define this deformation. One
is ℓ, a fundamental length, the other the sign ǫ5 .
The algebra ℜℓ,∞ is seen to be the algebra of the pseudo-Euclidean groups
E(1, 4) or E(2, 3), depending on whether ǫ5 is −1 or +1. It has a simple representation by differential operators in a five-dimensional space with coordinates
(ξ0 , ξ1 , ξ2 , ξ3 , ξ4 )
Pµ
Mµν
xµ
ℑ
=
=
=
=
i ∂ξ∂µ + iDPµ
i(ξµ ∂ξ∂ν − ξν ∂ξ∂µ ) + Σµν
ξµ + iℓ(ξµ ∂ξ∂ 4 − ǫ5 ξ 4 ∂ξ∂µ ) + ℓΣµ4
1 + iℓ ∂ξ∂ 4 + iℓDξ4
(3)
The set (Σµν , Σµ4 ) is an internal spin operator for the groups O(4, 1) (if ǫ5 = −1)
or O(3, 2) (if ǫ5 = +1) and DPµ and Dξ4 are derivations operating in the space
where (Σµν , Σµ4 ) acts. For practical calculations, in particular for the construction of quantum fields, it may be convenient to use this representation.
Notice however that only the Poincar´e part of E(1, 4) or E(2, 3) corresponds to
symmetry operations and only this part has to be implemented by unitary operators. Also, although an extra dimension is used in the representation space,
the space-time coordinates are still only four, noncommutative ones. Physical
consequences of the non-commutative space-time structure implied by the ℜℓ,∞
algebra have already been explored in a series of publications [6]-[11]. Depending on the sign of ǫ5 the time (ǫ5 = +1) or one space variable (ǫ5 = −1) will have
discrete spectrum. In any case ℓ, a new fundamental constant, sets a natural
scale for time and length. If ℓ is of the order of Planck’s length, observation
of most of the effects worked out in the cited references will be beyond present
experimental capabilities. However, if ℓ is much larger than Planck´s length
(for example of order 10−27 − 10−26 seconds) the effects might already be observable in the laboratory or in astrophysical observations. Some of the most
noteworthy effects arise from the modification of the phase space volume and
from interference effects.
However, most of the consequences worked out in the references [6]-[10] are
rather conservative, in the sense that they simply explore the nonvanishing of the
right-hand-side of the commutators of previously commuting variables. Deeper
consequences are to be expected from the radical change from a commutative to
a non-commutative space-time geometry, in particular from the new differential
algebra associated to the geometry. One such consequence will be described in
this paper. The new geometry was studied in Ref.[12] to which I will refer for
details and notation.
3
2
The differential algebra and an extended Dirac
equation
In the framework of the non-commutative geometry implied by the deformed
algebra, the differential algebra may be constructed either by duality from the
derivations of the algebra or from the triple (H, π(Uℜ ), D), where Uℜ is the
enveloping algebra of ℜℓ,∞ , to which a unit and, for later convenience, the
inverse of ℑ, are added.
Uℜ = {xµ , Mµν , pµ , ℑ, ℑ−1 , 1}
(4)
π(Uℜ ) is a representation of the Uℜ algebra in the Hilbert space H and D
is the Dirac operator, the commutator with the Dirac operator being used to
generate the one-forms. In a general non-commutative framework [13] [14] it
is not always possible to use the derivations of the algebra to construct by
duality the differential forms. In particular, many algebras have no derivations
at all. However when the algebra has enough derivations it is useful to consider
them [15] [16] because the correspondence of the non-commutative geometry
notions to the classical ones becomes very clear. One considers here the set V
of derivations with basis {∂µ , ∂4 } defined as follows1
∂µ (xν )
∂4 (xµ )
∂σ (Mµν )
∂µ (pν )
∂4 (Mµν )
=
=
=
=
=
ηµν ℑ
−ǫ5 ℓpµ ℑ
ησµ pν − ησν pµ
∂µ (ℑ) = ∂µ (1) = 0
∂4 (pµ ) = ∂4 (ℑ) = ∂4 (1) = 0
(5)
In the commutative (ℓ = 0) case a basis for 1-forms is obtained, by duality, from
the set {∂µ }. In the ℓ 6= 0 case the set of derivations {∂µ , ∂4 } is the minimal
set that contains the usual ∂µ ’s, is maximal abelian and is action closed on
the coordinate operators, in the sense that the action of ∂µ on xν leads to the
operator ℑ associated to ∂4 and conversely.
The operators that are associated to the physical coordinates are just the
four xµ , µ ∈ (0, 1, 2, 3). However, an additional degree of freedom appears in
the set of derivations. This is not a conjectured extra dimension but simply a
mathematical consequence of the algebraic structure of ℜℓ,∞ which, in turn, was
a consequence of the stabilizing deformation of relativistic quantum mechanics.
No extra dimension appears in the set of physical coordinates, because it does
not correspond to any operator in ℜℓ,∞ . However the derivations in V introduce, by duality, an additional degree of freedom in the exterior algebra. For
example, all quantum fields that are Lie algebra-valued connections will pick up
additional components. These additional components, in quantum fields that
are connections, are a consequence of the length parameter ℓ which does not
depend on its magnitude, but only on ℓ being 6= 0.
1 Notice
that the definition of ∂4 here is slightly different from the one in Ref.[12].
4
The Dirac operator [12] is
D = iγ a ∂a
(6)
with ∂a = (∂µ , ∂4 ), the γ’s being a basis for the Clifford algebras C (3, 2) or
C (4, 1)
γ 0 , γ 1 , γ 2 , γ 3, γ 4 = γ 5 ǫ5 = +1
(7)
γ 0 , γ 1 , γ 2 , γ 3 , γ 4 = iγ 5
ǫ5 = −1
How to construct quantum, scalar, spinor and gauge fields, as operators in Uℜ ,
has been described in [12]. In particular the role of the additional dimension
in the exterior algebra, on gauge interactions, has been emphasized (see also
[10]). Here another potentially interesting consequence for spinor fields will be
described. Because
h
i
− i k xν ,ℑ−1 }
− i k xν ,ℑ−1 }
+
+
pµ , e 2 ν {
= kµ e 2 ν {
(8)
a spinor field is written
Z
o
n
i
kν {xν ,ℑ−1 }
− i k xν ,ℑ−1 }
+ + d∗ v e 2
+
Ψ = d4 kδ(k 2 − m2 ) bk uk e 2 ν {
k k
Ψ ∈ Uℜ : DΨ − mΨ = 0
(9)
(10)
From the field a wave function is constructed operating on a vacuum state
ψ = Ψ |0i
(11)
Notice that both bk , d∗k and the elements of Uℜ operate on |0i, in particular
pµ |0i = 0. Now, for a massless field, the (extended) Dirac equation becomes
(12)
Dψ = iγ a ∂a ψ = iγ µ ∂µ + iγ 4 ∂4 ψ = 0
Write
ψ=e
− 2i kν {xν ,ℑ−1 }
+
u (k)
From
∂µ e
∂4 e
− 2i kν {xν ,ℑ−1 }
+
− 2i kν {xν ,ℑ−1 }
+
=
−ikµ e
− 2i kν {xν ,ℑ−1 }
+
µ
=
−iǫ5 ℓ −k pµ +
one obtains, using (13), (8) and (11)
γ µ kµ − γ 5 ℓ 21 k 2 u (k) = 0
γ µ kµ + iγ 5 ℓ 21 k 2 u (k) = 0
Let ǫ5 = −1. Iterating (14)
1 2
2k
e
ǫ5 = +1
ǫ5 = −1
ℓ2 2 2
u (k) = 0
k
k −
4
2
5
− 2i kν {xν ,ℑ−1 }
(13)
+
(14)
(15)
This equation has two solutions, the massless solution (k 2 = 0) and another
one, of large mass (ℓ being small)
k2 =
4
ℓ2
(16)
For ǫ5 = +1 one would obtain k 2 = 0 and
k2 = −
4
ℓ2
(17)
a tachyonic large k 2 solution.
The solutions of the extended Dirac equation
for k 2 = 0 are the usual ones.
2
To find the nature of the solutions for k = ℓ42 , ǫ5 = −1 and +1, use a
Majorana imaginary representation for the gamma matrices
0
σ2
iσ1 0
0 σ2
; γ2 =
; γ1 =
γ0 =
−σ2 0
σ2 0
0 iσ1
−σ2 0
iσ3 0
(18)
; γ 5 = iγ 0 γ 1 γ 2 γ 3 =
γ3 =
0
σ2
0 iσ3
2.1
ǫ5 = −1, k 2 =
4
ℓ2
In the rest frame k = (m0 = ± 2ℓ , 0, 0, 0). The second equation in (14) leads to
±γ 0 + iγ 5 u = 0
a
u=
Positive energy m0 = 2ℓ
ia
a
u=
Negative energy m0 = − 2ℓ
−ia
where a is an arbitrary two-vector. The solutions of non-zero momentum are ob- tained by the application of a proper Lorentz transformation exp i 21 αµν {γ µ , γ ν }+ .
One has u∗ 6= u, hence these solutions are Dirac spinors.
2.2
ǫ5 = +1, k 2 = − ℓ42
Here one makes k = (0, 0, 0, 2ℓ ), obtaining
γ 3 − γ 5 u′ = 0
a
Making u′ =
, a and b being two-vectors, yields
b
(σ2 + iσ3 ) a
(σ2 − iσ3 ) b
6
= 0
= 0
meaning that a and b are independent two vectors
b1
a1
;b =
a=
−b1
a1
with a1 and b1 real numbers u′∗ = u and this tachyonic, large k 2 , solution is
a Majorana spinor.
As before, general solutions would be obtained by the action of a Lorentz
transformation.
3
Coupling the two solutions
In the previous section it was seen how the extended Dirac equation, following
from the exterior
algebra of noncommutative space-time, has both a massless
and a large k 2 solution, large if ℓ is small. If, for example, ℓ is in the 10−27 −
10−26 seconds range, M = 2ℓ would be of the order of 1TeV. If the two solutions
mix, one expects that the massless solution would acquire a small mass as in
the seesaw mechanism proposed for neutrinos. In the seesaw mechanism the
large mass (of right-handed neutrinos) is hypothesized to be obtained from the
lepton number violation scale at grand unification. Here the large mass arises
from an independent solution of the same extended equation.
Let us call u1 the zero mass solution and u2 the large mass solution. Then let,
us assume that they are coupled by interaction with a background scalar field
that acquires a nonzero vacuum expectation value φ = hφi + h, with Lagrangian
µ
42
µ
u2 + (gu1 (hφi + h) u2 + h.c.)
(19)
L = u1 iγ ∂µ u1 + u2 iγ ∂µ + γ
ℓ
leading to the equations of motion:
iγ µ ∂µ u1+ g (hφi + h) u2
iγ ∂µ + γ 4 2ℓ u2 + g ∗ (hφi + h∗ ) u1
µ
= 0
= 0
(20)
With 2ℓ large, one neglects the kinetic term in the last equation and obtains, in
leading order,
u2 ≃ − 2ℓ g ∗ hφi γ 5 u1
ǫ5 = +1
(21)
ǫ5 = −1
u2 ≃ i 2ℓ g ∗ hφi γ 5 u1
Substitution in the first equation of (20) yields
3.1
ǫ5 = −1
2
2 ℓ
iγ µ ∂µ + i |g| hφi γ 5 u1 ≃ 0
2
which has a small mass solution with
2
2
2 ℓ
2
k = |g| hφi
2
7
For k = ± |g|2 hφi2 2ℓ , 0, 0, 0 the solutions are
u1 =
a
±ia
a small mass Dirac spinor. a is an arbitrary two-vector.
3.2
ǫ5 = +1
2
2 ℓ
iγ µ ∂µ − |g| hφi γ 5 u′1 ≃ 0
2
which has a small k 2 solution with
2
2
2 ℓ
k = − |g| hφi
2
2
2
2
For k = 0, 0, 0, |g| hφi 2ℓ the solution is

a1
 a1 

u′1 = 
 b1 
−b1

a small k 2 tachyonic Majorana spinor.
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8
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